Introduction

The aim of this document is to explain the basics of logical reasoning,
and hopefully improve the overall quality of debate.

The Concise Oxford English Dictionary defines logic as "the science of
reasoning, proof, thinking, or inference." Logic will let you analyze an
argument or a piece of reasoning, and work out whether it is likely to be
correct or not. You don't need to know logic to argue, of course; but if you
know even a little, you'll find it easier to spot invalid arguments.

There are many kinds of logic, such as fuzzy logic and constructive logic;
they have different rules, and different strengths and weaknesses. This document
discusses simple Boolean logic, because it's commonplace and relatively easy to
understand. When people talk about something being 'logical', they usually mean
the type of logic described here.

Arguments

An argument is, to quote the Monty Python
sketch, "a connected series of statements to establish a definite
proposition."

Many types of argument exist; we will discuss the deductive
argument. Deductive arguments are generally viewed as the most precise and
the most persuasive; they provide conclusive proof of their conclusion, and are
either valid or invalid.

Deductive arguments have three stages: premises, inference, and conclusion.
However, before we can consider those stages in detail, we must discuss the
building blocks of a deductive argument: propositions.

Propositions

A proposition is a statement which is either true or false. The
proposition is the meaning of the statement, not the precise arrangement of
words used to convey that meaning.

For example, "There exists an even prime number greater than two" is a
proposition. (A false one, in this case.) "An even prime number greater than two
exists" is the same proposition, re-worded.

Unfortunately, it's very easy to unintentionally change the meaning of a
statement by rephrasing it. It's generally safer to consider the wording of a
proposition as significant.

It's possible to use formal linguistics to analyze and re-phrase a statement
without changing its meaning; but how to do so is outside the scope of this
document.

Premises

A deductive argument always requires a number of core assumptions. These are
called premises, and are the assumptions the argument is built on; or
to look at it another way, the reasons for accepting the argument. Premises are
only premises in the context of a particular argument; they might be conclusions
in other arguments, for example.

You should always state the premises of the argument explicitly; this is the
principle of audiatur et
altera pars. Failing to state your assumptions is often viewed as
suspicious, and will likely reduce the acceptance of your argument.

The premises of an argument are often introduced with words such as
"Assume...", "Since...", "Obviously..." and "Because...." It's a good idea to
get your opponent to agree with the premises of your argument before proceeding
any further.

The word "obviously" is also often viewed with suspicion. It occasionally
gets used to persuade people to accept false statements, rather than admit that
they don't understand why something is 'obvious'. So don't be afraid to question
statements which people tell you are 'obvious' -- when you've heard the
explanation you can always say something like "You're right, now that I think
about it that way, it is obvious."

Inference

Once the premises have been agreed, the argument proceeds via a step-by-step
process called inference.

In inference, you start with one or more propositions which have been
accepted; you then use those propositions to arrive at a new proposition. If the
inference is valid, that proposition should also be accepted. You can use the
new proposition for inference later on.

So initially, you can only infer things from the premises of the argument.
But as the argument proceeds, the number of statements available for inference
increases.

There are various kinds of valid inference - and also some invalid kinds,
which we'll look at later in this document. Inference steps are often identified
by phrases like "therefore..." or "...implies that..."

Conclusion

Hopefully you will arrive at a proposition which is the conclusion of the
argument - the result you are trying to prove. The conclusion is the result of
the final step of inference. It's only a conclusion in the context of a
particular argument; it could be a premise or assumption in another
argument.

The conclusion is said to be affirmed on the basis of the
premises, and the inference from them. This is a subtle point which deserves
further explanation.

Implication in detail

Clearly you can build a valid argument from true premises, and arrive at a
true conclusion. You can also build a valid argument from false premises, and
arrive at a false conclusion.

The tricky part is that you can start with false premises, proceed via valid
inference, and reach a true conclusion. For example:

We can summarize these results as a "truth
table" for implication. The symbol "=>" denotes implication; "A" is the
premise, "B" the conclusion. "T" and "F" represent true and false
respectively.

Truth Table for Implication

Premise

Conclusion

Inference

A

B

A => B

false

false

true

false

true

true

true

false

false

true

true

true

If the premises are false and the inference valid, the conclusion can be
true or false. (Lines 1 and 2.)

If the premises are true and the conclusion false, the inference must be
invalid. (Line 3.)

If the premises are true and the inference valid, the conclusion must be
true. (Line 4.)

So the fact that an argument is valid doesn't necessarily mean that its
conclusion holds -- it may have started from false premises.

If an argument is valid, and in addition it started from true premises, then
it is called a sound argument. A sound argument must arrive at a true
conclusion.

Example argument

Here's an example of an argument which is valid, and which may or may not be
sound:

Premise: Every event has a cause

Premise: The universe has a beginning

Premise: All beginnings involve an event

Inference: This implies that the beginning of the universe involved an
event

Inference: Therefore the beginning of the universe had a cause

Conclusion: The universe had a cause

The proposition in line 4 is inferred from lines 2 and 3. Line 1 is then
used, with the proposition derived in line 4, to infer a new proposition in line
5. The result of the inference in line 5 is then re-stated (in slightly
simplified form) as the conclusion.

Spotting arguments

Spotting an argument is harder than spotting premises or a conclusion. Lots
of people shower their writing with assertions, without ever producing anything
you might reasonably call an argument.

Sometimes arguments don't follow the pattern described above. For example,
people may state their conclusions first, and then justify them afterwards. This
is valid, but it can be a little confusing.

To make the situation worse, some statements look like arguments but aren't.
For example:

"If the Bible is accurate, Jesus must either have been insane, an evil
liar, or the Son of God."

That's not an argument; it's a conditional statement. It doesn't state the
premises necessary to support its conclusion, and even if you add those
assertions it suffers from a number of other flaws which are described in more
detail in the "Atheist
Arguments" document.

An argument is also not the same as an explanation. Suppose that you are
trying to argue that Albert Einstein believed in God, and say:

"Einstein made his famous statement 'God does not play dice' because of his
belief in God."

That may look like a relevant argument, but it's not; it's an explanation of
Einstein's statement. To see this, remember that a statement of the form "X
because Y" can be re-phrased as an equivalent statement, of the form "Y
therefore X." Doing so gives us:

"Einstein believed in God, therefore he made his famous statement 'God does
not play dice'.

Now it's clear that the statement, which looked like an argument, is actually
assuming the result which it is supposed to be proving, in order to explain the
Einstein quote.

Furthermore, Einstein did not believe in a personal God concerned with human
affairs -- again, see the "Atheist
Arguments" document.

Logical Fallacies

There are a number of common pitfalls to avoid when constructing a deductive
argument; they're known as fallacies. In everyday English, we refer
to many kinds of mistaken beliefs as fallacies; but in logic, the term has a
more specific meaning: a fallacy is a technical flaw which makes an argument
unsound or invalid.

(Note that you can criticize more than just the soundness of an argument.
Arguments are almost always presented with some specific purpose in mind -- and
the intent of the argument may also be worthy of criticism.)

Arguments which contain fallacies are described as fallacious.
They often appear valid and convincing; sometimes only close inspection reveals
the logical flaw.

Below is a list of some common fallacies, and also some rhetorical devices
often used in debate. The list isn't intended to be exhaustive; the hope is that
if you learn to recognize some of the more common fallacies, you'll be able to
avoid being fooled by them.

Description of Fallacies

In order to understand what a fallacy is, one must
understand what an argument is. Very briefly, an argument consists of one or
more premises and one conclusion. A premise is a statement (a sentence that is
either true or false) that is offered in support of the claim being made, which
is the conclusion (which is also a sentence that is either true or false).

There are two main types of arguments: deductive and
inductive. A deductive argument is an argument such that the premises provide
(or appear to provide) complete support for the conclusion. An inductive
argument is an argument such that the premises provide (or appear to provide)
some degree of support (but less than complete support) for the conclusion. If
the premises actually provide the required degree of support for the conclusion,
then the argument is a good one. A good deductive argument is known as a valid
argument and is such that if all its premises are true, then its conclusion must
be true. If all the argument is valid and actually has all true premises, then
it is known as a sound argument. If it is invalid or has one or more false
premises, it will be unsound. A good inductive argument is known as a strong (or
"cogent") inductive argument. It is such that if the premises are true, the
conclusion is likely to be true.

A fallacy is, very generally, an
error in reasoning. This differs from a factual error, which is simply being
wrong about the facts. To be more specific, a fallacy is an "argument" in which
the premises given for the conclusion do not provide the needed degree of
support. A deductive fallacy is a deductive argument that is invalid (it is such
that it could have all true premises and still have a false conclusion). An
inductive fallacy is less formal than a deductive fallacy. They are simply
"arguments" which appear to be inductive arguments, but the premises do not
provided enough support for the conclusion. In such cases, even if the premises
were true, the conclusion would not be more likely to be true.

Examples of Fallacies

Inductive
Argument

Premise 1: Most American cats are domestic house cats.
Premise 2: Bill is an American cat.
Conclusion: Bill is domestic house cat.

Factual Error

Columbus
is the capital of the United States.

Deductive
Fallacy

Premise 1: If
Portland
is the capital of Maine, then it
is in Maine.
Premise 2: Portland is in
Maine.
Conclusion: Portland is the capital
of Maine.
(Portland is in
Maine, but Augusta
is the capital. Portland
is the largest city in Maine,
though.)

Inductive
Fallacy

Premise 1: Having just arrived in
Ohio, I saw a white squirrel.
Conclusion: All Ohio Squirrels are white.
(While there are many, many squirrels in Ohio,
the white ones are very rare).

Accent

Definition

Emphasis is used to suggest a meaning different from the actual content of
the proposition.

Accent is a form of fallacy through shifting meaning. In this case, the
meaning is changed by altering which parts of a statement are emphasized

Be particularly wary of this fallacy on the net, where it's easy to misread
the emphasis of what's written.

Examples

"We should not speak ill of our
friends"

It would be illegal to give away Free
Beer!

The first mate, seeking revenge on the captain, wrote in his
journal, "The Captain was sober today." (He suggests, by his emphasis, that the
Captain is usually drunk.

Ad hoc

Definition

As mentioned earlier, there is a difference between argument and
explanation. If we're interested in establishing A, and B is offered as
evidence, the statement "A because B" is an argument. If we're trying to
establish the truth of B, then "A because B" is not an argument, it's an
explanation.

The Ad Hoc fallacy is to give an after-the-fact explanation which doesn't
apply to other situations. Often this ad hoc explanation will be dressed up to
look like an argument. For example, if we assume that God treats all people
equally, then the following is an ad hoc explanation:

Examples

"I was healed from cancer."
"Praise the Lord, then. He is your healer."
"So, will He heal others who have cancer?"
"Er... The ways of God are mysterious."

Affirmation of the consequent

Definition

Any
argument of the following form is invalid: If A then B B Therefore,
A

This is the converse of Denial of the
Antecedent.

This fallacy is an argument of the form "A implies B, B is true, therefore A
is true." To understand why it is a fallacy, examine the truth table for
implication given earlier.

Examples

"If the universe had been created by a supernatural being, we
would see order and organization everywhere. And we do see order, not
randomness -- so it's clear that the universe had a creator."

If I am in Calgary, then I am in Alberta. I am in
Alberta, thus, I am in Calgary. (Of course, even though the premises are
true, I might be in Edmonton, Alberta.)

If the mill were polluting the
river then we would see an increase in fish deaths. And fish deaths have
increased. Thus, the mill is polluting the river.

Rebuttal

Show that even
though the premises are true, the conclusion could be false. In general, show
that B might be a consequence of something other than A. For example, the
fish deaths might be caused by pesticide run-off, and not the
mill.

Amphiboly

Definition

An amphiboly occurs when the construction of a
sentence allows it to have two different meanings. The fallacy occurs when the premises used in an argument are ambiguous because
of careless or ungrammatical phrasing.

Examples

Last
night I shot a burglar in my pajamas.

The Oracle of Delphi told Croseus
that if he pursued the war he would destroy a mighty kingdom. (What the
Oracle did not mention was that the kingdom he destroyed would be his own.
Adapted from Heroditus, The Histories.)

Save soap
and waste paper.

Belief in God fills a much-needed gap.

Rebuttal

Identify the ambiguous phrase and show the two
possible interpretations.

Anecdotal evidence

Definition

One of the simplest fallacies is to rely on anecdotal evidence.

It's quite valid to use personal experience to illustrate a point; but such
anecdotes don't actually prove anything to anyone. Your friend may say he met
Elvis in the supermarket, but those who haven't had the same experience will
require more than your friend's anecdotal evidence to convince them.

Anecdotal evidence can seem very compelling, especially if the audience
wants to believe it. This is part of the explanation for urban legends;
stories which are verifiably false have been known to circulate as anecdotes for
years.

Examples

"There's abundant proof that God exists and is still performing
miracles today. Just last week I read about a girl who was dying of cancer.
Her whole family went to church and prayed for her, and she was
cured."

Circulus in demonstrando

Definition

This fallacy occurs if you assume as a premise the conclusion which you wish
to reach. Often, the proposition is rephrased so that the fallacy appears to be
a valid argument.

Examples

"Homosexuals must not be allowed to hold government office. Hence any
government official who is revealed to be a homosexual will lose his job.
Therefore homosexuals will do anything to hide their secret, and will be open
to blackmail. Therefore homosexuals cannot be allowed to hold government
office."

Note that the argument is entirely circular; the premise is the same as the
conclusion. An argument like the above has actually been cited as the reason for
the British Secret Services' official ban on homosexual employees.

Circular arguments are surprisingly common, unfortunately. If you've already
reached a particular conclusion once, it's easy to accidentally make it an
assertion when explaining your reasoning to someone else.

Converting a conditional

This fallacy is similar to the Affirmation of
the Consequent, but phrased as a conditional statement.

Definition

This fallacy is an argument of the form "If A then B, therefore if B then
A."

Examples

"If educational standards are lowered, the quality of argument seen on the
Internet worsens. So if we see the level of debate on the net get worse over
the next few years, we'll know that our educational standards are still
falling."

Denial of the antecedent

Definition

This fallacy is an argument of the form "A implies B, A is false, therefore B
is false." The truth table
for implication makes it clear why this is a fallacy.

Note that this fallacy is different from Non Causa Pro
Causa. That has the form "A implies B, A is false, therefore B is false",
where A does not in fact imply B at all. Here, the problem isn't that
the implication is invalid; rather it's that the falseness of A doesn't allow us
to deduce anything about B.

Any argument of the following form is
invalid: If A then B Not A Therefore, Not B

This is the converse of the fallacy of Affirmation of
the Consequent.

Examples

If you get
hit by a car when you are six then you will die young. But you were not hit
by a car when you were six. Thus you will not die young. (Of course, you
could be hit by a train at age seven.)

If I am in Calgary then I am
in Alberta. I am not in Calgary, thus, I am not in Alberta.

"If the God of the Bible appeared to me, personally, that would certainly
prove that Christianity was true. But God has never appeared to me, so the
Bible must be a work of fiction."

Rebuttal

Show
that even though the premises are true, the conclusion may be false. In
particular, show that the consequence B may occur even though A does not
occur.

Equivocation / Fallacy of four
terms

Definition

The same word
is used with two different meanings.

Equivocation occurs when a key word is used with two or more different
meanings in the same argument. For example:

"What could be more affordable than free software? But to make
sure that it remains free, that users can do what they like with it, we must
place a license on it to make sure that will always be freely
redistributable."

One way to avoid this fallacy is to choose your terminology carefully before
beginning the argument, and avoid words like "free" which have many
meanings.

Examples

Criminal actions are
illegal, and all murder trials are criminal actions, thus all murder trials
are illegal. (Here the term "criminal actions" is used with two different
meanings.

The sign said "fine for
parking here", and since it was fine, I parked there.

All
child-murderers are inhuman, thus, no child-murderer is human.

A plane is a carpenter's
tool, and the Boeing 737 is a place, hence the Boeing 737 is a carpenter's tool.

Rebuttal

Identify the word which
is used twice, then show that a definition which is appropriate for one use
of the word would not be appropriate for the second use.

Ignoratio elenchi / Irrelevant
conclusion

Definition

Definition: An argument
which purports to prove one thing instead proves a different
conclusion.

The fallacy of Irrelevant Conclusion consists of claiming that an argument
supports a particular conclusion when it is actually logically nothing to do
with that conclusion.

Sadly, these kinds of irrelevant arguments are often successful, because they
make people to view the supposed conclusion in a more favorable light.

Examples

You should support the new housing bill. We
can't continue to see people living in the streets; we must have cheaper
housing. (We may agree that housing s important even though we disagree with
the housing bill.)

I say we should support affirmative action. White
males have run the country for 500 years. They run most of government and
industry today. You can't deny that this sort of discrimination is
intolerable. (The author has proven that there is discrimination, but not
that affirmative action will end that discrimination.)

A Christian may begin by saying that he will argue that the
teachings of Christianity are undoubtedly true. If he then argues at length that
Christianity is of great help to many people, no matter how well he argues he
will not have shown that Christian teachings are true.

Rebuttal

Show that
the conclusion proved by the author is not the conclusion that the author set
out to prove.

Non causa pro causa

Definition

The fallacy of Non Causa Pro Causa occurs when something is identified as the
cause of an event, but it has not actually been shown to be the cause.
For example:

Examples

"I took an aspirin and prayed to God, and my headache disappeared. So God
cured me of the headache."

This is known as a false cause fallacy. Two specific forms of non causa pro
causa fallacy are the cum hoc ergo
propter hoc and post hoc ergo
propter hoc fallacies.

Cum hoc ergo propter hoc This fallacy is a special case of the more general non causa pro
causa.

This fallacy is similar to post hoc ergo
propter hoc. The fallacy is to assert that because two events occur
together, they must be causally related. It's a fallacy because it ignores other
factors that may be the cause(s) of the events.

"Literacy rates have steadily declined since the advent of television.
Clearly television viewing impedes learning."

Non sequitur

Definition

A non sequitur is an argument where the conclusion is drawn from premises
which aren't logically connected with it. For example:

Examples

"Since Egyptians did so much excavation to construct the pyramids, they
were well versed in paleontology."

(Non sequiturs are an important ingredient in a lot of humor. They're still
fallacies, though.)

Plurium interrogationum / Many questions

Definition

This fallacy occurs when someone demands a simple (or simplistic) answer to a
complex question.

Examples

"Are higher taxes an impediment to business or not? Yes or
no?"

Many questions, also known as complex question, presupposition, loaded
question, or plurium interrogationum (Latin, "of many questions"), is a
logical fallacy. It is committed when someone asks a question that presupposes
something that has not been proven or accepted by all the people involved. This
fallacy is often used rhetorically, so that the question limits direct replies
to those that serve the questioner's agenda.

Reification / Hypostatization

Definition

Reification occurs when an abstract concept is treated as a concrete
thing.

Examples

"I noticed you described him as 'evil'. Where does this 'evil' exist within
the brain? You can't show it to me, so I claim it doesn't exist, and no man is
'evil'."

Prejudicial
Language

Definition

Loaded or emotive terms are used to attach value or
moral goodness to believing the proposition.

Examples

Right thinking
Canadians will agree with me that we should have another free vote on capital
punishment.

A reasonable person would agree that our income statement
is too low.

Senator Turner claims that the new tax rate will
reduce the deficit. (Here, the use of "claims" implies that what Turner
says is false.)

The proposal is likely to be resisted by the
bureaucrats on Parliament Hill. (Compare this to: The proposal is
likely to be rejected by officials on Parliament Hill.)

The animals were set free by animal
liberation terrorists.

Rebuttal

Identify
the prejudicial terms used (eg. "Right thinking Canadians" or "A reasonable
person"). Show that disagreeing with the conclusion does not make a person
"wrong thinking" or "unreasonable".

Anonymous Authorities

Definition

The authority
in question is not named. This is a type of appeal to authority because when
an authority is not named it is impossible to confirm that the authority is
an expert. However the fallacy is so common it deserves
special mention. A variation on this fallacy is the appeal to rumour.
Because the source of a rumour is typically not known, it is not possible
to determine whether to believe the rumour. Very often false and harmful
rumours are deliberately started in order to discredit an
opponent.

Examples

A government official said today that the new gun
law will be proposed tomorrow.

Experts agree that the best way to
prevent nuclear war is to prepare for it.

It is held that there are
more than two million needless operations conducted every year.

Rumour has it that the Prime Minster will declare another holiday in
October.

The university said the lab was
about to make a brilliant discovery when the ALF ruined the experiments.

Rebuttal

Argue that because we don't know the source of
the information we have no way to evaluate the reliability of
the information.

Style Over Substance

Definition

The
manner in which an argument (or arguer) is presented is taken to affect the
likelihood that the conclusion is true.

Examples

Nixon lost the
presidential debate because of the sweat on his forehead.

Trudeau
knows how to move a crowd. He must be right.

Why don't you take the
advice of that nicely dressed young man?

Rebuttal

While it is true that the
manner in which an argument is presented will affect whether people believe
that its conclusion is true, nonetheless, the truth of the conclusion does
not depend on the manner in which the argument is presented. In order to show
that this fallacy is being committed, show that the style in this case does
not affect the truth or falsity of the conclusion.

The style over substance fallacy occurs when one emphasises the way in which
the argument is presented, while marginalising (or outright ignoring) the
content of the argument.

False Analogy

Definition

In an
analogy, two objects (or events), A and B are shown to be similar. Then it is
argued that since A has property P, so also B must have property P. An
analogy fails when the two objects, A and B, are different in a way which
affects whether they both have property P.

Examples

Employees are
like nails. Just as nails must be hit in the head in order to make them work,
so must employees.

Government is like business, so just as business must
be sensitive primarily to the bottom line, so also must government. (But
the objectives of government and business are completely different, so
probably they will have to meet different criteria.)

Rebuttal

Identify the
two objects or events being compared and the property which both are said to
possess. Show that the two objects are different in a way which will affect
whether they both have that property.

Slothful Induction

Definition

The proper conclusion of an
inductive argument is denied despite the evidence to the
contrary.

Examples

Hugo has had twelve accidents n the last six months,
yet he insists that it is just a coincidence and not his
fault. (Inductively, the evidence is overwhelming that it is his
fault.)

Poll after poll
shows that the N.D.P will win fewer than ten seats in Parliament. Yet the
party leader insists that the party is doing much better than the polls
suggest. (The N.D.P. in fact got nine seats.)

Rebuttal

About all you can do
in such a case is to point to the strength of the inference.

Fallacy of Exclusion

Definition

Important evidence which would
undermine an inductive argument is excluded from consideration. The
requirement that all relevant information be included is called
the "principle of total evidence".

Examples

Jones is Albertan, and
most Albertans vote Tory, so Jones will probably vote Tory. (The information
left out is that Jones lives in Edmonton, and that most people in
Edmonton vote Liberal or N.D.P.)

The Leafs will probably win this
game because they've won nine out of their last ten. (Eight of the Leafs'
wins came over last place teams, and today they are playing the
first place team.)

Rebuttal

Give the missing evidence and show that it
changes the outcome of the inductive argument. Note that it is
not sufficient simply to show that not all of the evidence was included;
it must be shown that the missing evidence will change the conclusion.

Accident

Definition

A general rule is applied when circumstances suggest
that an exception to the rule should apply.

Examples

The law says
that you should not travel faster than 50 kph, thus even though your father
could not breathe, you should not have travelled faster than 50 kph.

It is good to return things you have borrowed. Therefore, you should return
this automatic rifle from the madman you borrowed it from. (Adapted from
Plato's Republic, Book I).

Rebuttal

Identify the generalization in question and
show that it s not a universal generalization. Then show that the
circumstances of this case suggest that the generalization ought not to
apply.

Genuine but
Insignificant Cause

Definition

The object or event identified as the cause
of an effect is a genuine cause, but insignificant when compared to the
other causes of that event. Note that this fallacy does not apply when all
other contributing causes are equally insignificant. Thus, it is not
a fallacy to say that you helped cause defeat the Tory government because
you voted Reform, for your vote had as much weight as any other vote, and
hence is equally a part of the cause.

Examples

Smoking is causing
air pollution in Edmonton. (True, but the effect of smoking is insignificant
compared to the effect of auto exhaust.)

By leaving your oven on
overnight you are contributing to global warming.

Rebuttal

Identify the much
more significant cause.

Complex Cause

Definition

The effect is caused by a number
of objects or events, of which the cause identified is only a part. A
variation of this is the feedback loop where the effect is itself a part of
the cause.

Examples

The accident was caused by the poor location of the
bush. (True, but it wouldn't have occurred had the driver not been drunk
and the pedestrian not been jaywalking.)

The Challenger explosion was
caused by the cold weather. (True, however, it would not have occurred had
the O-rings been properly constructed.)

People are in fear because
of increased crime. (True, but this has lead people to break the law as a
consequence of their fear, which increases crime even more.)

Rebuttal

Show
that all of the causes, and not just the one mentioned, are required to
produce the effect.

Inconsistency

Definition

The author asserts more than one
proposition such that the propositions cannot all be true. In such a case,
the propositions may be contradictories or they may
be contraries.

Examples

Montreal is about 200 km from Ottawa, while
Toronto is 400 km from Ottawa. Toronto is closer to Ottawa
than Montreal.

John is taller than Jake, and Jake is taller than
Fred, while Fred is taller than John.

Rebuttal

Assume that one of the
statements is true, and then use it as a premise to show that one of the
other statements is false.

Affirming the consequent is a logical fallacy that assumes that because a
hypothetical situation would bear a certain effect, that the occurrence of said
effect implies that the aforementioned situation occurred.

Appeal to consequences, also known as argumentum ad consequentiam, is
an argument that concludes a premise (typically a belief) to be either true or
false based on whether the premise leads to desirable or undesirable
consequences.

Argumentum ad nauseam or argument from repetition or argumentum ad
infinitum is a flawed argument, whereby some statement is made repeatedly
(possibly by different people) until nobody cares to refute it anymore, at which
point the statement is asserted to be true because it is no longer
challenged.

Also argumentum ad logicam - assumes that if an argument is
fallacious, its conclusion must be false.

This is the "fallacy fallacy" of arguing that a proposition is false because it has been presented as the conclusion of a fallacious argument. Remember always that fallacious arguments can arrive at true conclusions.

"Take the fraction 16/64. Now, cancelling a six on top and a six on the bottom, we get that 16/64 = 1/4."

The argument from ignorance, also known as argumentum ad ignorantiam
or argument by lack of imagination, is a logical fallacy in which it is claimed
that a premise is true only because it has not been proven false, or that a
premise is false only because it has not been proven true.

The argument from silence (also called argumentum a silentio in Latin)
is that the silence of a speaker or writer about X proves or suggests that the
speaker or writer is either ignorant of X or has a motive to remain silent about
X. When used as a logical proof in pure reasoning, the argument is classed among
the fallacies, but it may be valid circumstantial evidence in practical
reasoning.

Bulverism is a logical fallacy coined by C. S. Lewis where rather than
proving that an argument is wrong, a person instead assumes it wrong, and then
goes on to explain why the other person held that argument.

Denying the antecedent, a logical fallacy that assumes that because a
hypothetical situation would bear a certain effect, that the absence of the
hypothesised trigger situation means that the effect earlier described did not
occur.

The fallacy of the single cause, also known as joint effect or causal
oversimplification, is a logical fallacy of causation that occurs when it is
assumed that there is one, simple cause of an outcome when in reality it may
have been caused by a number of only jointly sufficient causes.

Ignoratio elenchi (also known as irrelevant conclusion) is the logical
fallacy of presenting an argument that may in itself be valid, but which proves
or supports a different proposition than the one it is purporting to prove or
support.

The perfect solution fallacy is a logical fallacy that occurs when an
argument assumes that a perfect solution exists and/or that a solution should be
rejected because some part of the problem would still exist after it was
implemented.

The practice of quoting out of context, sometimes referred to as contextomy,
is a logical fallacy and type of false attribution in which a passage is removed
from its surrounding matter in such a way as to distort its intended
meaning.